Relational operators in natural language

Purpose of relational operators

To say things such as "the name of the owner", we want to string together multiple relations (viz. name and owner). Relational operators allow us to make such statements.

Meaning

The meaning of relational operators $\smallsmile$ and $;$ is best explained by means of examples.

Assume we have a relation, label[Contract*Colour], which contains the colour of labels on contracts. A fact "1834" label "blue" means that contract 1834 has a blue label.

Also assume another relation stored[Contract*Location], which gives the location where a contract is stored. Fact "1834" store "cabinet 42" means that contract 1834 is stored in cabinet 42.

Converse

A relation can be altered by swapping the elements of every pair in the relation. Mathematically, $(a, b)$ is a different from $(b,a)$. In natural language, however, the meaning does not change. So if"1834" label "blue" means that contract 1834 has a blue label, "blue" label~ "1834" also means that contract 1834 has a blue label.

• The sentence: "All contracts with a blue label are stored in cabinet 42." is represented as "blue" (label\stored) "cabinet 42". Literally it says: For every contract, if it has a blue label, then it is stored in cabinet 42.

Composition

The sentence "A contract with a blue label is stored in cabinet 42." can be represented as "blue" (label~;stored) "cabinet 42". Literally it says: There is a contract that has a blue label and is stored in cabinet 42.

Natural language templates

There is a pattern to this. A computer can generate a literal translation from the formula to natural language. However, that translation looks clumsy, verbose and elaborate. It is up to you to turn that in normal language. For examples click here. The systematic translation is given in the following table:

Formally	Natural language template
a (r;s) b	There exists an x : if a r xthen x s b.
b r~ a	a r b.

The natural language translation for b r~ ais the same as language translation for a r b.

Other explanation

Would you like a different explanation of the relational operators? This page explains the relational operators in terms of set theory. An explanation in logic is given here. Click here for some algebraic rules about relational operators.